The Soliton-Ricci Flow over Compact Manifolds

We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to a Perelman's modified backward Ricci type flow with some special restriction conditions. The restriction conditions are motivated by convexity results for Perelman's $\mathcal{W}$-functional over convex subsets inside adequate subspaces of Riemannian metrics. We show indeed that the Soliton-Ricci flow is generated by the gradient flow of the restriction of Perelman's $\mathcal{W}$-functional over such subspaces. Assuming long time existence of the Soliton-Ricci flow we show exponentially fast convergence to a shrinking Ricci soliton provided that the Bakry-Emery-Ricci tensor is uniformly strictly positive with respect to the evolving metric.